Conformal gradient vector fields on Riemannian manifolds with boundary

TL;DR

This paper proves that certain conformal gradient vector fields on compact Riemannian manifolds with boundary imply the manifold is isometric to a hemisphere of the sphere, under conditions on Ricci curvature, scalar curvature, and energy.

Contribution

It establishes new rigidity results linking conformal vector fields and geometric conditions to the manifold being a hemisphere of the sphere.

Findings

Manifolds with nontrivial conformal gradient vector fields and controlled Ricci curvature are isometric to hemispheres.

Einstein manifolds with nonzero conformal gradient vector fields have positive scalar curvature and are hemispheres.

Manifolds with constant scalar curvature and conformal vector fields have positive scalar curvature and are hemispheres.

Abstract

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$ to be isometric to a hemisphere of $\mathbb{S}^{n}$. We also prove that if an Einstein manifold admits nonzero conformal gradient vector field, then its scalar curvature is positive and it is isometric to a hemisphere of $\mathbb{S}^{n}$. Furthermore, we prove that if $ M $ admits a nontrivial conformal vector field and has constant scalar curvature, then the scalar curvature is positive. Finally, a suitable control on the energy of a conformal vector field implies that $M$ is isometric to a hemisphere $\mathbb{S}^n_+$.